Quantum computers can be perfectly simulated by classical computers (but only inefficiently, as far as we know). So at best, you could claim that computation in the brain may involve quantum effects and therefore it would not be tractable to simulate it on a classical computer. But a quantum computer is still a Turing machine.

Of course, we can not yet rule out the possibility that there exist more powerful models of computation than Turing machines. We just know quantum effects don't lead there.